Apparently I have quite a few thoughts on the matter
--
At the moment the I'm the nearest catalyst that I can think of that would cause that change would be the mission to wave, especially if the return of that mission is followed up with Airi getting swept up in a bandit situation.

Sasuke "Dying" would shake Kakashi as one of his students was nearly lost but he would recover, as he does. But if while he is recovering from that shock Airi get's placed in danger? Something is going to shift it would have to as Kakashi isn't one to breakdown, add in Guy not leaping at the chance to rescue her for some reason, nothing malicious, something like injury on his team that is grounding them or confidence in any of the chunin teams being able to get it taken care of swiftly. something that would a reasonable excuse to a jonin, that just doesn't resonate with Kakashi who is just getting back from a surprise A rank with his team.

That would firmly switch things from 'this is mostly fine' to 'THIS IS NOT OKAY AT ALL'

We could also have a discussion between Airi and Kakashi before the mission to wave to add emphasis to the change in emotions, reasons why they are there could be simple as that they are using the backyard for a chakra control training day, using the leaf exercise or something equally as unimpactful on the surroundings or Kakashi is walking Naruto there as he coaches the boy on something that is hindering him.

As for things slightly more immediate, the only thing I can think of would be Airi following Kakashi to Rin's grave and learning a bit about his guilt. [url= https://vignette.wikia.nocookie.net/naruto/images/b/bc/Rin_Nohara.png/revision/latest?cb=20150805145941]Rin[/url] was a member of his chunin team but despite her dying in the war she is not on the memorial stone. The issue I have with that is that it would likely just become an exposition dump.